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        Questions tagged [lie-groups]

        Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

        4
        votes
        0answers
        174 views

        Gram-Schmidt map as a Riemannian submersion

        We equip $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{O}(n)$ with their left-invariant metrics, whose restrictions to the corresponding neutral elements is the standard inner product $\mathrm{Trace}(AB^{\...
        4
        votes
        3answers
        158 views

        Real points of reductive groups and connected components

        Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...
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        votes
        1answer
        127 views

        An easier reference than “On the Functional Equations Satisfied by Eisenstein Series”?

        I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands. http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...
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        vote
        0answers
        71 views

        Basic notation question involving Lie Groups and Lie algebras

        I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/...
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        votes
        0answers
        23 views

        Compact image of adjoint action

        Let $G$ be a connected Lie group and $H$ a connected Lie-subgroup. Suppose that for every compact subset $K\subset G/H$ the $H$-orbit $H.K$ is relatively compact in $G/H$. Is it true that the image ...
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        votes
        3answers
        280 views

        $SO(m+1)$-equivariant maps from $S^m$ to $S^m$

        Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$. Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$. Question 1: Is F the ...
        2
        votes
        1answer
        102 views

        Simple modules for direct sum of simple Lie algebras

        I think that the following statement is true, but I do not know how to prove it. Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
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        vote
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        Homotopy classes of maps between special unitary Lie group. Correction [duplicate]

        An hour ago I asked a question (under the same title) but I used a wrong notation. Here is the improved version. We consider a special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and ...
        5
        votes
        1answer
        201 views

        Homotopy classes of maps between special unitary Lie groups

        I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now. We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...
        1
        vote
        1answer
        185 views

        Homotopy of group actions

        Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...
        2
        votes
        0answers
        135 views

        More general form of Fourier inversion formula

        My question begins as follows: Suppose $G$ is a compact (Lie) group and $A$ is a $G$-module, and denote the action of $G$ on $A$ by $\alpha$. Fix $a\in A$ and view $$ f:g\mapsto \alpha_g(a) $$ as an $...
        5
        votes
        1answer
        154 views

        Definition of a Dirac operator

        So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as: $D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
        3
        votes
        1answer
        88 views

        A converse of Cartan's automatic continuity theorem

        Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...
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        votes
        0answers
        109 views

        Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

        Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups: $$ 1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1 $$ There exists a ...
        3
        votes
        1answer
        203 views

        Extensions of compact Lie groups

        Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups $$ 0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0, $$ $$ 0\rightarrow G\...

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        山西福彩快乐十分钟
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